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MATH 1009 TEST 3 - SOLUTIONS March 2011

1. [10 Marks] Find the derivative of the given functions:

[3] (a) () = 25 + 3.

[2] (b) () = (ln )2, > 0.

[5] (c) () =(

32

1 )4

.

Solution:

(a) () = 22 ln 2 (5) + (3)4 = (5 ln 2) 25 34.

(b) () = 2 ln

1

=2

ln .

(c) () = 4(

32

1 )3

(

32

1 )

= 4(

32

1 )3

(

6(1 ) 32(1)(1 )2

)=

= 4(

32

1 )3

6 32

(1 )2(

= 4 33 6

(1 )3 3(2 )(1 )2 =

3247(2 )(1 )5

)

2. [ 6 Marks] Use implicit differentiation to find

:

23 + 32 = 2.

Solution:

Differentiate both sides with respect to :

2 3 + 2 32

+ 6 = 0.

Solve for

:

=6 23

322.

3. [ 12 Marks] The quantity demanded each week is related to the unit price by thedemand equation

= () =

360 4 (0 < < 90).The elasticity of demand is given by the formula () =

()

().

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[9] (a) Is the demand elastic or inelastic at = 70? at = 50?[2] (b) For what price is the demand unitary?[1] (c) If the unit price is increased slightly from 70, will the revenue increase or decrease?

Solution:(a) We compute

() =(

360 4)

=1

2(

360 4)

1

2 (4) = 2360 4 .

Thus, () = ()

()=

2360 4

360 4 =2

360 4 .

(70) =2 70

360

4

70

=7

4> 1, so the demand is elastic when = 70.

(50) =2 50

360 4 50 =100

160< 1, so the demand is inelastic when = 50.

(b) () = ()

()=

2

360 4 = 1 when 2 = 3604, so for = 60 the demand is unitary.

(c) Since the demand is elastic when = 70, the revenue will decrease if the unit price isincreased slightly from 70.

4. [ 12 Marks] Find and classify the critical numbers of the following functions:

(a)[5] () = 3 12 + 74. (b)[7] () = 2

2 4.

Solution:(a) () is a polynomial function, so its domain is the set of all real numbers. The only sourceof critical numbers is the condition () = 0.

() = 32 12 = 3(2 4) = 2( 2)( + 2) = 0 when = 2 or = 2.As () < 0 for

2 < < 2 and () > 0 otherwise, by the First Derivative Test, at =

2

the function has a local maximum and at = 2 - a local minimum.

(b) () is a rational function, defined on the set of all real numbers for which 2 4 = 0,that is, = 2.

() =2(2 4) 2(2)

(2 4)2 =22 8(2 4)2 =

2( 4)(2 4)2 .

The derivative () is not defined for = 2. However, since = 2 is not in the domain of ,the number = 2 is not a critical number of .

() = 0 when = 0 or = 4.

As () < 0 for 0 < < 4 and () > 0 otherwise, by the First Derivative Test, at = 0 thefunction has a local maximum and at = 4 - a local minimum.

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